Question

Find the general solution of the differential equation $\frac{dy}{dx}=\frac{x+1}{2-y},y\ne 2$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \frac{dy}{dx}=\frac{x+1}{2-y} \\ \Rightarrow \left(2-y\right)dy=\left(x+1\right)dx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \int \left(2-y\right)dy=\int \left(x+1\right)dx \\ \Rightarrow 2y-\frac{y^2}{2}=\frac{x^2}{2}+x+C_1 \\ \Rightarrow \frac{x^2}{2}+x+C_1-2y+\frac{y^2}{2}=0 \\ \Rightarrow x^2+2x+y^2+2C_1-4y=0 \\ \Rightarrow x^2+y^2+2x-4y+C=0\left[\mathrm{Where},C=2C_1\right] \end{gathered}$$